There exist several results on packings on Grassmannian spaces but they are rather centered in finding codes such that the mutual chordal distance between different elements \([{\bf X}_i],[{\bf X}_j]\)  is close to maximal. For example, in [1]} we find bounds for the mutual distance of any code with a fixed number of elements (this is known as the Rankin bound). Gilbert–Varshamov bounds have also been obtained for that chordal distance by resorting to calculations of the volume of a metric ball of radius \(\delta \)  in \(\mathbb {G}r(M,{\mathbb {C}}^{T})\) , see [2]} and [3]}. The case that \(\delta \)  is sufficiently small was analyzed in [3]}, [5]}, and the real case has also been studied, see [6]} and references therein. But to our knowledge our results are the first theoretical bounds on codes focusing on the explicit use of the chordal product, which is the key figure of merit in non–coherent communications.
